A numerical model of the renal distal tubule
Catherine
Lloyd
Auckland Bioengineering Institute, The University of Auckland
Model Status
This is the CellML description of Chang and Fujita's 1999 numerical model of the renal distal tubule. It should be noted that the initial conditions used in this description represent those in the early distal tubule. For model representations of the late distal tublule, these initial values should be replaced with those listed in the orginal paper for the late distal tubule. The model from the original paper has been modified slightly to include differential equations defining the change in solute concentrations over time. These equations were added to facilitate the use of the CellML code in CMISS. Note that the model is not running correctly in COR or OpenCell and there are unit inconsistencies.
Model Structure
ABSTRACT: A numerical model of the rat distal tubule was developed to simulate water and solute transport in this nephron segment. This model incorporates the following: 1) Na-Cl cotransporter, K-Cl cotransporter, Na channel, K channel, and Cl channel in the luminal membrane; 2) Na-K-ATPase, K channel, and Cl channel in the basolateral membrane; and 3) conductances for Na, K, and Cl in the paracellular pathway. Transport rates were calculated using kinetic equations. Axial heterogeneity was represented by partitioning the model into two subsegments with different sets of model parameters. Model equations derived from the principles of mass conservation and electrical neutrality were solved numerically. Values of the model parameters were adjusted to minimize a penalty function that was devised to quantify the difference between model predictions and experimental results. The developed model could simulate the water and solute transport of the distal tubule in the normal state, as well as in conditions including thiazide or amiloride application and various levels of sodium load and tubular flow rate.
The original paper reference is cited below:
A numerical model of the renal distal tubule, Hangil Chang and Toshiro Fujita, 1999,
American Journal of Physiology
, 276, F952-F959.
PubMed ID: 10362782
diagram of the model
Transport mechanisms of model tubule. In the luminal cell membrane, there are Na-Cl cotransporter, K-Cl cotransporter, Na channel, K channel, and Cl channel. In the basolateral cell membrane, there are Na-K-ATPase, K channel, and Cl channel. In the paracellular pathway, which faces luminal and basolateral compartments, there are conductances for sodium, potassium, and chloride. Axial heterogeneity of the distal tubule was represented by changing the model parameters in the early and the late parts of the model tubule.
$\frac{d \mathrm{C\_m\_Na}}{d \mathrm{time}}=-(\mathrm{J\_mc\_Na}+\mathrm{J\_ms\_Na})\frac{d \mathrm{C\_s\_Na}}{d \mathrm{time}}=\mathrm{J\_ms\_Na}-\mathrm{J\_sc\_Na}\frac{d \mathrm{C\_c\_Na}}{d \mathrm{time}}=\mathrm{J\_mc\_Na}+\mathrm{J\_sc\_Na}\frac{d \mathrm{C\_m\_K}}{d \mathrm{time}}=-(\mathrm{J\_mc\_K}+\mathrm{J\_ms\_K})\frac{d \mathrm{C\_s\_K}}{d \mathrm{time}}=\mathrm{J\_ms\_K}-\mathrm{J\_sc\_K}\frac{d \mathrm{C\_c\_K}}{d \mathrm{time}}=\mathrm{J\_mc\_K}+\mathrm{J\_sc\_K}\frac{d \mathrm{C\_m\_Cl}}{d \mathrm{time}}=-(\mathrm{J\_mc\_Cl}+\mathrm{J\_ms\_Cl})\frac{d \mathrm{C\_s\_Cl}}{d \mathrm{time}}=\mathrm{J\_ms\_Cl}-\mathrm{J\_sc\_Cl}\frac{d \mathrm{C\_c\_Cl}}{d \mathrm{time}}=\mathrm{J\_mc\_Cl}+\mathrm{J\_sc\_Cl}$
$\mathrm{J\_mc\_Na}=\mathrm{J\_mc\_NaCl}+\mathrm{G\_mc\_Na}\mathrm{G\_mc\_Na}=\mathrm{P\_mc\_Na}\frac{F(\mathrm{psi\_m}-\mathrm{psi\_c})}{\mathrm{RT}}\frac{\mathrm{C\_m\_Na}-\mathrm{C\_c\_Na}e^{-\left(\frac{F}{\mathrm{RT}}\right)(\mathrm{psi\_m}-\mathrm{psi\_c})}}{1.0-e^{-\left(\frac{F}{\mathrm{RT}}\right)(\mathrm{psi\_m}-\mathrm{psi\_c})}}\mathrm{J\_mc\_NaCl}=\mathrm{J\_mc\_NaCl\_max}\frac{\frac{\mathrm{C\_m\_Na}}{\mathrm{K\_mc\_Na\_NaCl}}\frac{\mathrm{C\_m\_Cl}}{\mathrm{K\_mc\_Cl\_NaCl}}-\frac{\mathrm{C\_c\_Na}}{\mathrm{K\_mc\_Na\_NaCl}}\frac{\mathrm{C\_c\_Cl}}{\mathrm{K\_mc\_Cl\_NaCl}}}{(1.0+\frac{\mathrm{C\_m\_Na}}{\mathrm{K\_mc\_Na\_NaCl}}\frac{\mathrm{C\_m\_Cl}}{\mathrm{K\_mc\_Cl\_NaCl}})(1.0+\frac{\mathrm{C\_c\_Na}}{\mathrm{K\_mc\_Na\_NaCl}})(1.0+\frac{\mathrm{C\_c\_Cl}}{\mathrm{K\_mc\_Cl\_NaCl}})+(1.0+\frac{\mathrm{C\_c\_Na}}{\mathrm{K\_mc\_Na\_NaCl}}\frac{\mathrm{C\_c\_Cl}}{\mathrm{K\_mc\_Cl\_NaCl}})(1.0+\frac{\mathrm{C\_m\_Na}}{\mathrm{K\_mc\_Na\_NaCl}})(1.0+\frac{\mathrm{C\_m\_Cl}}{\mathrm{K\_mc\_Cl\_NaCl}})}$
$\mathrm{J\_mc\_K}=\mathrm{J\_mc\_KCl}+\mathrm{G\_mc\_K}\mathrm{G\_mc\_K}=\mathrm{P\_mc\_K}\frac{F(\mathrm{psi\_m}-\mathrm{psi\_c})}{\mathrm{RT}}\frac{\mathrm{C\_m\_K}-\mathrm{C\_c\_K}e^{-\left(\frac{F}{\mathrm{RT}}\right)(\mathrm{psi\_m}-\mathrm{psi\_c})}}{1.0-e^{-\left(\frac{F}{\mathrm{RT}}\right)(\mathrm{psi\_m}-\mathrm{psi\_c})}}\mathrm{J\_mc\_KCl}=\mathrm{J\_mc\_KCl\_max}\frac{\frac{\mathrm{C\_m\_K}}{\mathrm{K\_mc\_K\_KCl}}\frac{\mathrm{C\_m\_Cl}}{\mathrm{K\_mc\_Cl\_KCl}}-\frac{\mathrm{C\_c\_K}}{\mathrm{K\_mc\_K\_KCl}}\frac{\mathrm{C\_c\_Cl}}{\mathrm{K\_mc\_Cl\_KCl}}}{(1.0+\frac{\mathrm{C\_m\_K}}{\mathrm{K\_mc\_K\_KCl}}\frac{\mathrm{C\_m\_Cl}}{\mathrm{K\_mc\_Cl\_KCl}})(1.0+\frac{\mathrm{C\_c\_K}}{\mathrm{K\_mc\_K\_KCl}})(1.0+\frac{\mathrm{C\_c\_Cl}}{\mathrm{K\_mc\_Cl\_KCl}})+(1.0+\frac{\mathrm{C\_c\_K}}{\mathrm{K\_mc\_K\_KCl}}\frac{\mathrm{C\_c\_Cl}}{\mathrm{K\_mc\_Cl\_KCl}})(1.0+\frac{\mathrm{C\_m\_K}}{\mathrm{K\_mc\_K\_KCl}})(1.0+\frac{\mathrm{C\_m\_Cl}}{\mathrm{K\_mc\_Cl\_KCl}})}$
$\mathrm{J\_mc\_Cl}=\mathrm{J\_mc\_NaCl}+\mathrm{J\_mc\_KCl}+\mathrm{G\_mc\_Cl}\mathrm{G\_mc\_Cl}=\mathrm{P\_mc\_Cl}\frac{-1.0F(\mathrm{psi\_m}-\mathrm{psi\_c})}{\mathrm{RT}}\frac{\mathrm{C\_m\_Cl}-\mathrm{C\_c\_Cl}e^{-\left(\frac{-1.0F}{\mathrm{RT}}\right)(\mathrm{psi\_m}-\mathrm{psi\_c})}}{1.0-e^{-\left(\frac{-1.0F}{\mathrm{RT}}\right)(\mathrm{psi\_m}-\mathrm{psi\_c})}}$
$\mathrm{J\_sc\_Na}=-3.0\mathrm{J\_a}\mathrm{J\_a}=\mathrm{J\_a\_max}\frac{1.0}{1.0+\left(\frac{\mathrm{K\_Na\_ATPase}}{\mathrm{C\_c\_Na}}\right)^{3.0}}$
$\mathrm{J\_sc\_K}=2.0\mathrm{J\_a}+\mathrm{G\_sc\_K}\mathrm{G\_sc\_K}=\mathrm{P\_sc\_K}\frac{F(\mathrm{psi\_s}-\mathrm{psi\_c})}{\mathrm{RT}}\frac{\mathrm{C\_s\_K}-\mathrm{C\_c\_K}e^{-\left(\frac{F}{\mathrm{RT}}\right)(\mathrm{psi\_s}-\mathrm{psi\_c})}}{1.0-e^{-\left(\frac{F}{\mathrm{RT}}\right)(\mathrm{psi\_s}-\mathrm{psi\_c})}}$
$\mathrm{J\_sc\_Cl}=\mathrm{G\_sc\_Cl}\mathrm{G\_sc\_Cl}=\mathrm{P\_sc\_Cl}\frac{-1.0F(\mathrm{psi\_s}-\mathrm{psi\_c})}{\mathrm{RT}}\frac{\mathrm{C\_s\_Cl}-\mathrm{C\_c\_Cl}e^{-\left(\frac{-1.0F}{\mathrm{RT}}\right)(\mathrm{psi\_s}-\mathrm{psi\_c})}}{1.0-e^{-\left(\frac{-1.0F}{\mathrm{RT}}\right)(\mathrm{psi\_s}-\mathrm{psi\_c})}}$
$\mathrm{J\_ms\_Na}=\mathrm{G\_ms\_Na}\mathrm{G\_ms\_Na}=\mathrm{P\_ms\_Na}\frac{F(\mathrm{psi\_m}-\mathrm{psi\_s})}{\mathrm{RT}}\frac{\mathrm{C\_m\_Na}-\mathrm{C\_s\_Na}e^{-\left(\frac{F}{\mathrm{RT}}\right)(\mathrm{psi\_m}-\mathrm{psi\_s})}}{1.0-e^{-\left(\frac{F}{\mathrm{RT}}\right)(\mathrm{psi\_m}-\mathrm{psi\_s})}}$
$\mathrm{J\_ms\_K}=\mathrm{G\_ms\_K}\mathrm{G\_ms\_K}=\mathrm{P\_ms\_K}\frac{F(\mathrm{psi\_m}-\mathrm{psi\_s})}{\mathrm{RT}}\frac{\mathrm{C\_m\_K}-\mathrm{C\_s\_K}e^{-\left(\frac{F}{\mathrm{RT}}\right)(\mathrm{psi\_m}-\mathrm{psi\_s})}}{1.0-e^{-\left(\frac{F}{\mathrm{RT}}\right)(\mathrm{psi\_m}-\mathrm{psi\_s})}}$
$\mathrm{J\_ms\_Cl}=\mathrm{G\_ms\_Cl}\mathrm{G\_ms\_Cl}=\mathrm{P\_ms\_Cl}\frac{-1.0F(\mathrm{psi\_m}-\mathrm{psi\_s})}{\mathrm{RT}}\frac{\mathrm{C\_m\_Cl}-\mathrm{C\_s\_Cl}e^{-\left(\frac{-1.0F}{\mathrm{RT}}\right)(\mathrm{psi\_m}-\mathrm{psi\_s})}}{1.0-e^{-\left(\frac{-1.0F}{\mathrm{RT}}\right)(\mathrm{psi\_m}-\mathrm{psi\_s})}}$
$\mathrm{J\_Na}=\mathrm{J\_mc\_Na}+\mathrm{J\_ms\_Na}$
$\mathrm{J\_K}=\mathrm{J\_mc\_K}+\mathrm{J\_ms\_K}$
$\mathrm{J\_Cl}=\mathrm{J\_mc\_Cl}+\mathrm{J\_ms\_Cl}$
$\mathrm{Osm\_m}=\mathrm{C\_m\_Na}+\mathrm{C\_m\_K}+\mathrm{C\_m\_Cl}+\mathrm{C\_m\_Imp}\mathrm{Osm\_c}=\mathrm{C\_c\_Na}+\mathrm{C\_c\_K}+\mathrm{C\_c\_Cl}+\mathrm{C\_c\_Imp}\mathrm{Osm\_s}=\mathrm{C\_s\_Na}+\mathrm{C\_s\_K}+\mathrm{C\_s\_Cl}+\mathrm{C\_s\_Imp}$
$\mathrm{J\_mc\_v}=\mathrm{L\_mc\_v}\mathrm{RT}(\mathrm{Osm\_m}-\mathrm{Osm\_c})$
$\mathrm{J\_ms\_v}=\mathrm{L\_ms\_v}\mathrm{RT}(\mathrm{Osm\_m}-\mathrm{Osm\_s})$
$\mathrm{J\_sc\_v}=\mathrm{L\_sc\_v}\mathrm{RT}(\mathrm{Osm\_s}-\mathrm{Osm\_c})$
$\mathrm{J\_v}=\mathrm{J\_mc\_v}+\mathrm{J\_ms\_v}$
Concentration of chloride in the cytosol compartment
Concentration of sodium in the cytosol compartment
Na+ flow though Na-Cl cotransporter across apical cell membrane
Flux of Na+ through Na-Cl cotransporter from luminal compartment to cytosol compartment across apical cell membrane
Concentration of sodium in the cytosol compartment
Concentration of chloride in the cytosol compartment
water and mineral metabolism
A numerical model of the renal distal tubule
A numerical model of the rat distal tubule was developed to simulate water and solute transport in this nephron segment. This model incorporates the following: 1) Na-Cl cotransporter, K-Cl cotransporter, Na channel, K channel, and Cl channel in the luminal membrane; 2) Na-K-ATPase, K channel, and Cl channel in the basolateral membrane; and 3) conductances for Na, K, and Cl in the paracellular pathway. Transport rates were calculated using kinetic equations. Axial heterogeneity was represented by partitioning the model into two subsegments with different sets of model parameters. Model equations derived from the principles of mass conservation and electrical neutrality were solved numerically. Values of the model parameters were adjusted to minimize a penalty function that was devised to quantify the difference between model predictions and experimental results. The developed model could simulate the water and solute transport of the distal tubule in the normal state, as well as in conditions including thiazide or amiloride application and various levels of sodium load and tubular flow rate.
transporters
ion channels
TOSHIRO FUJITA
HANGIL CHANG
distal tubule
Concentration of potassium in the cytosol compartment
Cl- flow through Na-Cl cotransporter and K-Cl cotransporter
Flux of Cl- through Na-Cl cotransporter and K-Cl cotransporter from luminal compartment to cytosol compartment across apical cell membrane
Concentration of chloride in the luminal compartment
K+ flow through K-Cl across apical cell membrane
Flux of K+ through K-Cl cotransporter from luminal compartment to cytosol compartment across apical cell membrane
Na+ flow through Na channel across apical cell membrane
Flux of Na+ through Na channel from luminal compartment to cytosol compartment across apical cell membrane
Concentration of sodium in the luminal compartment
Flux of Na+ through Na-Cl cotransporter from luminal compartment to cytosol compartment across apical cell membrane
Concentration of sodium in the tissue fluid compartment
K+ flow through Na-K-ATPase across basolateral cell membrane
Flux of K+ through Na-K-ATPase from tissue fluid compartment to cytosol compartment across basolateral cell membrane
Concentration of chloride in the tissue fluid compartment
Concentration of potassium in the cytosol compartment
Concentration of chloride in the tissue fluid compartment
Concentration of chloride in the cytosol compartment
Cl- flow through Cl diffusive channel across paracellular pathway
Flux of Cl- through Cl diffusive channel from luminal compartment to tissue fluid compartment across paracellular pathway
Concentration of chloride in the luminal compartment
Concentration of potassium in the luminal compartment
Concentration of potassium in the tissue fluid compartment
Concentration of chloride in the tissue fluid compartment
Concentration of sodium in the tissue fluid compartment
Concentration of chloride in the cytosol compartment
Concentration of chloride in the tissue fluid compartment
Concentration of chloride in the luminal compartment
Concentration of potassium in the luminal compartment
Cl- flow through Cl channel across basolateral cell membrane
Flux of Cl- through Cl channel from tissue fluid compartment to cytosol compartment across basolateral cell membrane
Concentration of potassium in the luminal compartment
Concentration of potassium in the tissue fluid compartment
Flux of K+ through Na-K-ATPase from tissue fluid compartment to cytosol compartment across basolateral cell membrane
Na+ flow through Na diffusive channel across paracellular pathway
Flux of Na+ through Na diffusive channel from luminal compartment to tissue fluid compartment across paracellular pathway
Concentration of potassium in the cytosol compartment
Concentration of sodium in the luminal compartment
Flux of Na+ through Na-Cl cotransporter from luminal compartment to cytosol compartment across apical cell membrane
Concentration of potassium in the tissue fluid compartment
Concentration of chloride in the luminal compartment
Concentration of sodium in the luminal compartment
Flux of K+ through K-Cl cotransporter from luminal compartment to cytosol compartment across apical cell membrane
Concentration of potassium in the cytosol compartment
Flux of Cl- through Na-Cl cotransporter and K-Cl cotransporter from luminal compartment to cytosol compartment across apical cell membrane
Concentration of chloride in the luminal compartment
Concentration of chloride in the cytosol compartment
Concentration of potassium in the tissue fluid compartment
K+ flow through K diffusive channel across paracellular pathway
Flux of K+ through K diffusive channel from luminal compartment to tissue fluid compartment across paracellular pathway
K+ flow through K channel across apical cell membrane
Flux of K+ through K channel from luminal compartment to cytosol compartment across apical cell membrane
Na+ flow through Na-K-ATPase across basolateral compartment
Flux of Na+ through Na-K-ATPase from cytosol compartment to tissue fluid compartment across basolateral cell membrane
K+ flow through K channel across basolateral cell membrane
Flux of K+ through K channel from tissue fluid compartment to cytosol compartment across basolateral cell membrane
Concentration of chloride in the luminal compartment
Flux of K+ through K-Cl cotransporter from luminal compartment to cytosol compartment across apical cell membrane
Flux of Cl- through Na-Cl cotransporter and K-Cl cotransporter from luminal compartment to cytosol compartment across apical cell membrane
Concentration of sodium in the cytosol compartment
Concentration of potassium in the luminal compartment
Concentration of sodium in the tissue fluid compartment
Cl- flow through Cl channel across apical cell membrane
Flux of Cl- through Cl channel from luminal compartment to cytosol compartment across apical cell membrane
Concentration of sodium in the luminal compartment
Concentration of chloride in the cytosol compartment
Concentration of sodium in the cytosol compartment
Flux of Na+ through Na-K-ATPase from cytosol compartment to tissue fluid compartment across basolateral cell membrane